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E-raamat: Harmonic and Subharmonic Function Theory on the Hyperbolic Ball

(University of South Carolina)
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Harmonic and Subharmonic Function Theory on the Hyperbolic BallSuurem pilt
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This detailed and comprehensive treatment is ideal for established researchers in the field as well as graduate students who wish to learn more about harmonic and subharmonic function theory on the hyperbolic ball and upper half-space. The only prerequisites are a standard beginning graduate course in real analysis.

This comprehensive monograph is ideal for established researchers in the field and also graduate students who wish to learn more about the subject. The text is made accessible to a broad audience as it does not require any knowledge of Lie groups and only a limited knowledge of differential geometry. The author's primary emphasis is on potential theory on the hyperbolic ball, but many other relevant results for the hyperbolic upper half-space are included both in the text and in the end-of-chapter exercises. These exercises expand on the topics covered in the chapter and involve routine computations and inequalities not included in the text. The book also includes some open problems, which may be a source for potential research projects.

Arvustused

'The author gives a comprehensive treatment of invariant potential theory. The exposition is clear and elementary. This book is recommended to graduate students and researchers interested in this field. It is a very good addition to the mathematical literature.' Hiroaki Aikawa, MathSciNet

Muu info

A detailed treatment of potential theory on the real hyperbolic ball and half-space aimed at researchers and graduate students.
Preface xi
1 Mobius Transformations
1(5)
1.1 Notation
1(1)
1.2 Inversion in Spheres and Planes
2(2)
1.3 Mobius Transformations
4(2)
2 Mobius Self-Maps of the Unit Ball
6(11)
2.1 Mobius Transformations of B
6(3)
2.2 The Hyperbolic Metric on B
9(3)
2.3 Hyperbolic Half-Space H
12(3)
2.4 Exercises
15(2)
3 The Invariant Laplacian, Gradient, and Measure
17(14)
3.1 The Invariant Laplacian and Gradient
17(2)
3.2 The Fundamental Solution of Δh
19(2)
3.3 The Invariant Measure on B
21(3)
3.4 The Invariant Convolution on B
24(3)
3.5 Exercises
27(4)
4 H-Harmonic and H-Subharmonic Functions
31(28)
4.1 The Invariant Mean-Value Property
31(4)
4.2 The Special Case n = 2
35(2)
4.3 H-Subharmonic Functions
37(4)
4.4 Properties of H-Subharmonic Functions
41(4)
4.5 Approximation by C∞ H-Subharmonic Functions
45(3)
4.6 The Weak Laplacian and Riesz Measure
48(3)
4.7 Quasi-Nearly H-Subharmonic Functions
51(5)
4.8 Exercises
56(3)
5 The Poisson Kernel and Poisson Integrals
59(23)
5.1 The Poisson Kernel for Δh
59(3)
5.2 Relationship between the Euclidean and Hyperbolic Poisson Kernel
62(2)
5.3 The Dirichlet Problem for B
64(4)
5.4 The Dirichlet Problem for Br
68(2)
5.5 Eigenfunctions of Δh
70(6)
5.6 The Poisson Kernel on H
76(2)
5.7 Exercises
78(4)
6 Spherical Harmonic Expansions
82(14)
6.1 Dirichlet Problem for Spherical Harmonics
83(3)
6.2 Zonal Harmonic Expansion of the Poisson Kernel
86(4)
6.3 Spherical Harmonic Expansion of H-Harmonic Functions
90(4)
6.4 Exercises
94(2)
7 Hardy-Type Spaces of H-Subharmonic Functions
96(18)
7.1 A Poisson Integral Formula for Functions in Hp, 1 ≤ p ≤ ∞
97(4)
7.2 Completeness of Hp, 0 < p ≤ ∞
101(2)
7.3 H-Harmonic Majorants for H-Subharmonic Functions
103(6)
7.4 Hardy--Orlicz Spaces of H-Subharmonic Functions
109(3)
7.5 Exercises
112(2)
8 Boundary Behavior of Poisson Integrals
114(25)
8.1 Maximal Functions
114(6)
8.2 Non-tangential and Radial Maximal Function
120(5)
8.3 Fatou's Theorem
125(2)
8.4 A Local Fatou Theorem for H-Harmonic Functions
127(4)
8.5 An Lp Inequality for Mα ƒ for 0 < p ≤ 1
131(3)
8.6 Example
134(3)
8.7 Exercises
137(2)
9 The Riesz Decomposition Theorem for H-Subharmonic Functions
139(34)
9.1 The Riesz Decomposition Theorem
140(3)
9.2 Applications of the Riesz Decomposition Theorem
143(6)
9.3 Integrability of H-Superharmonic Functions
149(6)
9.4 Boundary Limits of Green Potentials
155(7)
9.5 Non-tangential Limits of H-Subharmonic Functions
162(7)
9.6 Exercises
169(4)
10 Bergman and Dirichlet Spaces of H-Harmonic Functions
173(43)
10.1 Properties of Dργ and Bργ
174(4)
10.2 Mobius Invariant Spaces
178(2)
10.3 Equivalence of Bργ and Dργ for y > (n -- 1)
180(6)
10.4 Integrability of Functions in Bργ and Dργ
186(7)
10.5 Integrability of Eigenfunctions of Δh
193(5)
10.6 Three Theorems of Hardy and Littlewood
198(7)
10.7 Littlewood-Paley Inequalities
205(6)
10.8 Exercises
211(5)
References 216(5)
Index of Symbols 221(2)
Index 223
Manfred Stoll is Distinguished Professor Emeritus in the Department of Mathematics at the University of South Carolina. His books include Invariant Potential Theory in the Unit Ball of Cn (Cambridge, 1994) and Introduction to Real Analysis (1997).
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